Three children return home after playing outside, and their father tells them that at least one of them has a muddy face. He repeats the phrase “Step forward if you have a muddy face” until all and only the children with muddy faces have stepped forward.

If there’s only one child with a muddy face, then she’ll step forward immediately — she can see that no other children have muddy faces, so her father must be talking about her. Each of the other children will see her muddy face and stand fast, since they have no way of knowing whether their own faces are muddy.

If there are two children with muddy faces, then no one will step forward after the first request, since each might think the father is addressing the other one. But when no one steps forward after the first request, each will realize that there must be two children with muddy faces, and that she herself must be one of them. So both will step forward after the second request, and the rest will stand fast.

A pattern emerges: If there are n children with muddy faces, then n will step forward after the nth request.

But now imagine a scenario in which more than one of the children has a muddy face, but the father does not tell them that at least one of them has a muddy face. Now no one steps forward after the first request, for the same reason as before. But no one steps forward at the second request either, because the fact that no one stepped forward after the first request no longer means that there is more than one child with a muddy face.

This is perplexing. In the second scenario all the children can see that at least one of them has a muddy face, so it seems needless for the father to tell them so. But without his statement the argument never gets going; despite his repeated requests, no child will ever step forward. What’s missing?

(From Michael Clark, Paradoxes From A to Z, 2007.)

The map of the continental United States contains an elf making chicken.

He’s known as Mimal, after the states that make him up: Minnesota (hat), Iowa (head), Missouri (shirt), Arkansas (pants), and Louisiana (boots).

Fittingly, the chicken is Kentucky and the tin pan is Tennessee.

A study in perspective by University of Hertfordshire psychologist Richard Wiseman:

(Thanks, Paul.)

In early July 1836, three boys searching for rabbits’ burrows near Edinburgh came upon some thin sheets of slate set into the side of a cliff. On removing them, they discovered the entrance to a little cave, where they found 17 tiny coffins containing miniature wooden figures.

According to the Scotsman‘s account later that month, each of the coffins “contained a miniature figure of the human form cut out in wood, the faces in particular being pretty well executed. They were dressed from head to foot in cotton clothes, and decently laid out with a mimic representation of all the funereal trappings which usually form the last habiliments of the dead. The coffins are about three or four inches in length, regularly shaped, and cut out from a single piece of wood, with the exception of the lids, which are nailed down with wire sprigs or common brass pins. The lid and sides of each are profusely studded with ornaments, formed with small pieces of tin, and inserted in the wood with great care and regularity.”

Some accounts say that the coffins had been laid in tiers, the lower appearing decayed and the topmost quite recent, but Edinburgh University historian Allen Simpson believes that all were placed in the niche after 1830, about five years before the boys discovered them.

Who placed them there, and why, remain mysterious. Simpson suggests that they may be an attempt to provide a decent symbolic burial for the victims of murderers William Burke and William Hare, who had sold 17 corpses to local doctor Robert Knox in 1828 for use in anatomy lessons. But 12 of Burke and Hare’s victims were women, and the occupants of the fairy coffins are all dressed as men.

So investigations continue. The eight surviving coffins and their tiny occupants are on display today at the National Museum of Scotland.

For more than 500 million years something has been making hexagonal burrows on the floor of the deep sea. Each network of tiny holes leads to a system of tunnels under the surface. The creature that makes them, known as Paleodictyon nodosum, has never been discovered. It might be a worm or perhaps a protist; the structure might be its means of farming its own food or the remains of a nest for protecting eggs. Fossils have been found in the limestone of Nevada and Mexico, and the burrows even turn up in the drawings of Leonardo da Vinci. But what makes them, and how, remain a mystery.

Somewhat related: When puzzling screw-shaped structures (below) were unearthed in Nebraska in the 1890s they were known as “devil’s corkscrews” and attributed to freshwater sponges or some sort of coiling plant. They were finally recognized as the burrows of prehistoric beavers only when a fossilized specimen, Palaeocastor, was found inside one.

(Thanks, Paul.)

When we say that the function of the heart is to pump the blood, what do we mean, exactly? Typically an object’s function is something that confers some good or contributes to some goal: In pumping blood my heart keeps me alive; in grasping objects my hands help me manipulate my environment.

But is that right? Suppose someone designs a sewing machine with a self-destruct button. Pressing the button will never have good consequences for anyone, and no one will ever set a goal that’s furthered by blowing up the machine. Still, it seems correct to say that the button’s function is to destroy the machine.

Another example, from Johns Hopkins philosopher Peter Achinstein: “Suppose that a magnificent chair was designed as a throne for the king, i.e., it was designed to seat the king. However, it is actually used by the king’s guards to block a doorway in the palace. Finally, suppose that although the guards attempt to block the doorway by means of that chair they are unsuccessful. The chair is so beautiful that it draws crowds to the palace to view it, and people walk through the doorway all around the chair to gaze at it. But its drawing such crowds does have the beneficial effect of inducing more financial contributions for the upkeep of the palace, although this was not something intended. What is the function of this chair?”

(Peter Achinstein, “Function Statements,” Philosophy of Science, September 1977.)

The following remarkable coincidence will be read with interest: Sometime since it was announced that a man at Titusville, Pennsylvania, committed suicide for the strange reason that he had discovered that he was his own grandfather. Leaving a dying statement explaining this singular circumstance, we will not attempt to unravel it, but give his own explanation of the mixed-up condition of his kinsfolk in his own words. He says, ‘I married a widow who had a grown-up daughter. My father visited our house very often, fell in love with my stepdaughter, and married her. So my father became my son-in law, and my step-daughter my mother, because she was my father’s wife. Some time afterwards, my wife gave birth to a son; he was my father’s brother-in-law, and my uncle, for he was the brother of my step-mother. My father’s wife — i.e. my step-daughter — had also a son; he was, of course, my brother, and in the mean time my grandchild, for he was the son of my daughter. My wife was my grandmother, because she was my mother’s mother. I was my wife’s husband and the grandchild at the same time. And as the husband of a person’s grandmother is his grandfather, I was my own grandfather.’ After this logical conclusion, we are not surprised that the unfortunate man should have taken refuge in oblivion. It was the most married family and the worst mixed that we ever heard of. To unravel such an entangling alliance could not have resulted otherwise than in an aberration of mind and subsequent suicide.

— Littell’s Living Age, May 9, 1868

(Yes, I know about the song!) (Thanks, Dave.)